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REPORTS ON MATHEMATICAL LOGIC 47 (2012), 125–145 DOI:10.4467/20842589RM.12.006.0687 George VOUTSADAKIS CATEGORICAL ABSTRACT ALGEBRAIC LOGIC: COORDINATIZATION IS ALGEBRAIZATION A b s t r a c t. The methods of categorical abstract algebraic logic are employed to show that the classical process of the coordinatization of abstract (affine plane) geometry can be viewed under the light of the algebraization of logical systems. This link offers, on the one hand, a new perspective to the coordinatization of geometry and, on the other, enriches abstract algebraic logic by bringing under its wings a very well-known geometric process, not known hitherto to be related or amenable to its methods and techniques. The algebraization takes the form of a deductive equivalence between two institutions, one corresponding to affine plane geometry and the other to Hall ternary rings. Received 6 June 2011 Keywords: Abstract Geometry, Affine Plane Geometry, Coordinatization, Hall Ternary Rings, Institutions, Algebraic Institutions, Deductive Equivalence, Categorical Abstract Algebraic Logic, Interpretations, Equivalent Algebraic Semantics 2010 AMS Subject Classification: Primary: 03G27 Secondary: 18A15, 20N10, 51A25 126 GEORGE VOUTSADAKIS 1. Introduction Around the end of the nineteenth and the beginning of the last century, several authors considered the process of coordinatization of various axiomatically defined abstract geometries. Among them were the affine plane geometry, affine plane geometry with the Desargues and Pappus properties, projective plane geometry etc. A classic introductory book to be consulted on this topic, which partly inspired our investigations and to which the reader will be referred for several constructions and results that will be used in the present paper, is Blumenthal’s “A Modern View of Geometry” [3]. A more recent reference on this material, that is now available online in its second edition, is Peter Cameron’s “Projective and Polar Spaces” [6]. Finally, a general reference pertaining to the topic that may prove of further help to the interested reader is the handbook [5]. The general process of coordinatization consists, generally speaking, of using the points and incidence structure of a given abstract geometry as a basis for constructing an abstract algebraic structure. Ordinarily, the points of the geometry serve as the building blocks for the universe of the algebraic structure. Then, the structural properties of the geometry are used to define one or more algebraic operations on this universe. These operations are shown to satisfy several algebraic properties that may lead to a characterization of the resulting algebraic structure. These algebraic structures are then used to assign coordinates to the points of the original abstract geometry. One of the benefits of this process of coordinatization is that, having the algebraic structure of the coordinates at hand, enables one to manipulate the points and lines of the geometry and study several of their properties by algebraic means, using the operations of the resulting algebras. For instance, one may define slopes, equations of lines, discover intersection points by solving systems of equations etc. Taking the opposite point of view, another significant advantage of this process is that, given an abstract algebra of the kind used in the coordinatization, there may be a reverse construction for obtaining an abstract geometry by defining points and lines using the elements of the universe of the algebraic structure. Then, by studying the structure of the resulting geometry, and taking into account the interplay between algebra and geometry, one may develop useful geometric intuitions concerning aspects of the original algebraic structures. So, as it turns out, on several occasions, coor- COORDINATIZATION IS ALGEBRAIZATION 127 dinatization has positive consequences not only in the study of the original abstract geometry, by making it amenable to algebraic techniques, but also in the study of a class of abstract algebras, by possibly endowing their operations with some geometric dimension that may make their presentation more intuitive and, thus, better understood and more easily and efficiently studied. The initial motivation for the attempt to relate the coordinatization of geometry to the algebraization of logical systems arose from the fact that the two processes are very similar in intent and nature. As with the coordinatization of geometry, the goal of abstract algebraic logic, as crystalized in the seminal memoirs monograph “Algebraizable Logics” of Blok and Pigozzi [2], is to associate with a logical system a class of abstract algebras in such a way that the logical entailment of the system may be studied by taking advantage of the algebraic properties of the class of algebras. More precisely, the logical entailment is translated to the equational entailment of the corresponding class of algebras and, then, its study may be carried out by a detailed analysis of the structure of the corresponding algebraic congruences. This can be successfully carried out under the condition that a close enough relationship exists between the original logical and the induced algebraic entailment systems. Thus, leaving the apparent discrepancies in the details involved aside, there are essential similarities in the intention and the goals of coordinatization in geometry and algebraization in logic: both strive to associate with their objects of study (geometries and logics, respectively) an algebraic system or a class of algebraic systems, in such a way that the basic structures under consideration (incidence and consequence, respectively) are reflected in, or can be expressed in terms of, the algebraic properties of the associated algebraic systems. Then, by looking at the algebras (algebraic operations and equational consequence, respectively) one can draw conclusions about the original geometry or logic. The passage to abstract algebra in both cases is assumed to facilitate the study of the original objects, since the structure of algebraic objects is, in many instances, better understood and more readily analyzable than that of the original objects. In this paper, we carry out this project of relating the coordinatization of abstract (affine plane) geometry, as presented in [3], with the process of algebraization of logics expressed as institutions, as it was developed in the original papers on this subject [15, 16]. The subfield of categorical abstract 128 GEORGE VOUTSADAKIS algebraic logic (CAAL), whose definitions and techniques we employ in this work, is the one responsible for making the original concepts, methods and techniques employed in abstract algebraic logic (see, e.g., [9, 7, 10]) to logics expressed as institutions [11, 12] or π-institutions [8]. In the sense of CAAL, the algebraization of an institution [16] takes the form of a deductive equivalence [15] of the institution with another institution whose classes of models consist of algebraic systems and whose sentences consist of equations. Thus, the way coordinatization is related to algebraization consists, roughly speaking, of expressing the abstract geometry as a logical system in the form of an institution, and, then, using the coordinatizing abstract algebraic structures as algebraic models of another algebraic institution. A critical selection of sentences of the geometric institution and of equations of the algebraic institution must be made, so that the coordinatization process may be accurately captured by the process of establishing a deductive equivalence between the geometric institution and the corresponding algebraic institution. This process involves the construction of translations between incidence relations and equations and vice-versa, which become interpretations between entailments on incidence relations and equational entailments, that are inverses of one another in the precise sense stipulated by CAAL (see [2] and [15]). The link established in this way between the coordinatization of abstract geometry and the general process of algebraization of institutional logics is interesting for several reasons. First, it reveals a link between two seemingly unrelated processes: that of associating a class of algebras as an algebraic semantics to a logical system and that of associating a class of algebras as coordinate algebras to the models of an abstract geometry. Second, it sheds new light to the nature of coordinatization by linking it, for instance, to the association of Boolean algebras to classical propositional calculus through this very abstract channel. Finally, it enriches abstract algebraic logic by bringing under its umbrella a very well-known subfield of geometric investigations, not known previously to be amenable to its methods and techniques. Thus, the main formal result, Theorem 16, of our work, that spans both geometry and logic, justifies (at least to some extent) the motto: “The coordinatization of abstract geometry is a form of algebraization, in the formal sense attributed to this term by the modern theory of abstract algebraic logic.” COORDINATIZATION IS ALGEBRAIZATION 129 The paper is organized as follows: In Section 2, abstract geometries, the geometries of affine planes, are introduced and morphisms between them, termed geometric morphisms, are defined. In Section 3, an account of the process of the coordinatization of the affine plane, as detailed in, e.g., [3], is presented.1 Moreover, coordinate systems are added to abstract geometries and the resulting coordinated abstract geometries and coordinated geometric morphisms between them are formally defined. The resulting category is denoted by AG. In Section 4 our main work starts in earnest. An institution AG is built that formalizes the system of abstract (affine plane) geometry as a logical system. This involves the construction of its syntax, consisting of formulas made up using the incidence relation symbol of the geometry, and of its semantics, which consists of the models of the axioms of affine plane geometry. The two interact through the satisfaction relation, which, as is expected, involves an incidence formula being true if, intuitively speaking, the corresponding intended incidence relation holds in the chosen geometric model. In Section 5, the properties of the coordinatizing rings of the abstract geometries proven in [3] are revisited together with the process by which, given such a ring, called a Hall ternary ring, an abstract geometry may be constructed. If that geometry is endowed with a canonically associated coordinate system, then the resulting coordinate ring coincides with the originally given Hall ternary ring. In Section 6, the institution GA of geometric algebra is constructed. It is essentially the institution of the equational theory of Hall ternary rings. The fact that Hall ternary rings provide the coordinate rings for abstract geometry is the connecting link between this institution and that of abstract geometry, introduced in Section 4. Indeed, in Section 7, it is shown that GA is an algebraic semantics for AG by exhibiting an interpretation from the sentences of AG into the equations of GA. Finally, in Section 8, it is shown that an inverse interpretation from the equations of GA into the sentences of AG exists. The existence of these two mutually inverse interpretations establish the fact that AG is algebraizable in the sense of CAAL with GA as its equivalent algebraic semantics. Since AG is the institution of affine plane geomeries whereas its equivalent algebraic semantics GA is the institution 1 A referee has pointed out that the technical details of the treatment of the algebraization of abstract geometry might become less tedious if a Tarski-Szmielew style presentation of affine plane geometry [14], using points as the only primitive objects, is chosen in place of the presentation used in [3], employing both points and lines. 130 GEORGE VOUTSADAKIS of the coordinate rings of affine plane geometries, the deductive equivalence established between these two institutions reveals the close relationship between coordinatization in geometry and algebraization in abstract algebraic logic that was displayed in the boxed motto above. For all unexplained categorical terminology and notation the reader is encouraged to consult any of the standard references [1, 4, 13]. 2. Abstract Geometry and Geometric Morphisms The notion of abstract geometry (see, e.g., Chapter IV of [3], where this affine plane geometry is defined) abstracts the very basic features common in many applied plane geometries, in particular those of many finite geometries and of the Euclidean plane geometry. An abstract geometry G = hP, L, Ii consists of a set P of abstract points, a set L of abstract lines, and an incidence relation I ⊆ P × L, which are subject to three postulates (the term parallel is used to characterize two lines l, l′ ∈ L, when there does not exist any point p ∈ P incident to both): Postulate 1 If p0 , p1 ∈ P with p0 6= p1 , then there exists a unique line l ∈ L incident to both p0 and p1 ; Postulate 2 If p ∈ P and l ∈ L, such that p is not incident to l, then there exists a unique line l′ parallel to l, such that p is incident to l′ ; Postulate 3 There exists at least one quadruple of distinct points in P , no three of which are incident to the same line. Given two abstract geometries G = hP, L, Ii and G′ = hP ′ , L′ , I ′ i, a morphism of abstract geometries (or a geometric morphism, for short) f : G → G′ is a pair f = hf0 , f1 i, such that f0 : P → P ′ and f1 : L → L′ , preserving the incidence relations, i.e., satisfying: pI l implies f0 (p) I ′ f1 (l). A geometric morphism f : G → G′ is said to be strict if the implication above is an equivalence. It is shown in Theorem IV.1.2 of [3] that, given an abstract geometry G = hP, L, Ii, every line l ∈ L of G is incident to the same number2 n of 2 possibly an infinite cardinal COORDINATIZATION IS ALGEBRAIZATION 131 points and, also, in Corollary IV.1.1 of [3], that every point p ∈ P of G is incident to the same number n + 1 of lines, i.e., one more than the number of points incident with any line of G. Moreover, the parallel class of any one line l ∈ L of G (i.e., the set containing l together with all lines in L parallel to l) has exactly n members, as many as the number of points on any one line of the abstract geometry. 3. Coordinatization of an Abstract Geometry Given an abstract geometry G = hP, L, Ii, there is a well-known process that can be used to coordinatize its points. We briefly recall this process here, but the reader is referred to Chapter IV of [3] for many more details. By Postulate 3 of an abstract geometry, there exists at least one quadruple of distinct points, no three of which are incident with the same line. We select such a quadruple of points (O, I, X, Y ). The point O is referred to as the origin, I as the unit point, the unique line incident to O and X is called the x-line, the unique line incident to O and Y is called the y-line and the unique line incident to O and I is called the unit line. Next, an arbitrary set of elements R with the same cardinality n as that of the set of points on the unit line is picked and a one-to-one correspondence ρ between the points on the unit line and the elements of R is established. Labels are assigned to the points of R in such a way that two labels 0 and 1 are reserved for the images of the origin O and the unit point I, respectively, i.e., ρ(O) = 0 and ρ(I) = 1. Once this selection has been fixed, a point p ∈ P can be assigned coordinates c(p) = (a, b) ∈ R2 in the following way: • If p is incident to the unit line, then c(p) = (ρ(p), ρ(p)). Thus, for instance, c(O) = (0, 0) and c(I) = (1, 1). • If p is not incident to the unit line, then, if the unique line incident to p and parallel to the y-line intersects the unit line at u, with c(u) = (a, a), and the unique line incident to p and parallel to the x-line intersects the unit line at v, with c(v) = (b, b), we set c(p) = (a, b). Furthermore, given a pair (a, b) ∈ R2 , it can be shown that there exists a unique point p ∈ P of G, such that c(p) = (a, b). We denote that point by p(a, b). Finally, given m, b ∈ R, it can be shown that there exists a unique 132 GEORGE VOUTSADAKIS line l ∈ L, with slope m and y-intercept b in a way very similar to the ordinary process used in the Euclidean plane. That line will be denoted by l(m, b). And, conversely, given a line l, not belonging to the same parallel class as the y-line, there exist unique m, b ∈ R, such that l has slope m and y-intercept b. We write d(l) = (m, b). Given an abstract geometry G = hP, L, Ii, we write S = h(O, I, X, Y ), R, ρi to denote the coordinate system of G, including the coordinate set R and the bijection ρ from the points incident to the unit line to R, described in this section. The pair hG, Si is called a coordinated abstract geometry. Let hG, Si and hG′ , S ′ i be two coordinated abstract geometries. A morphism of coordinated abstract geometries (or, more simply a coordinated geometric morphism) f¯ : hG, Si → hG′ , S ′ i is a pair f¯ = hf, f ∗ i, where f : G → G′ is a geometric morphism and f ∗ : R → R′ is a mapping between the corresponding coordinate sets, such that 1. f0 maps O, I, X, Y to O ′ , I ′ , X ′ , Y ′ , respectively; 2. f ∗ (ρ(p)) = ρ′ (f0 (p)), for all p incident to the unit line of hG, Si. The morphism f¯ is called strict if f : G → G′ is strict. Proposition 1. Let hG, Si and hG′ , S ′ i be two coordinated abstract geometries and f¯ : hG, Si → hG′ , S ′ i be a strict coordinated geometric morphism. Then 1. c′ (f0 (p)) = (f ∗ )2 (c(p)), for all p ∈ P , where (f ∗ )2 denotes coordinatewise application of f ∗ on an order pair; 2. d′ (f1 (l)) = (f ∗ )2 (d(l)), for all l ∈ L, not in the parallel class of the y-line. Proof. Suppose that c(p) = (a, b). Thus, the unique line through p that is parallel to OY intersects OI at p(a, a) and the unique line through p that is parallel to OX intersects OI at p(b, b). By the definition and the strictness of f¯, we get that p′ (f ∗ (a), f ∗ (a)) is the point of intersection of the line through f0 (p) that is parallel to the line O′ Y ′ and, similarly, that p′ (f ∗ (b), f ∗ (b)) is the point of intersection of the line through f0 (p) that is parallel to O ′ X ′ . Therefore, we obtain that c′ (f0 (p)) = (f ∗ (a), f ∗ (b)), as was to be shown. The second part may be proven similarly. COORDINATIZATION IS ALGEBRAIZATION 133 Proposition 1 shows that the entire strict coordinated geometric morphism f¯ can be reconstructed from knowledge of f ∗ and the pairs hG, Si and hG′ , S ′ i alone. Thus, in the sequel we will identify a strict coordinated geometric morphism f¯ = hf, f ∗ i with its second component f ∗ and express this by saying that f : hG, Si → hG′ , S ′ i is a strict coordinated geometric morphism, where f : R → R′ (i.e., we identify f¯ with f ∗ : R → R′ , but, then drop the ∗ from the notation). Moreover, since in what follows only strict coordinated geometric morphisms will enter our discussion, we will drop the qualifiers “strict coordinated” and use geometric morphism to refer to strict coordinated geometric morphisms. Let AG denote the category with objects all coordinated abstract geometries and morphisms all (strict coordinated) geometric morphisms between them. For the formulation of the institution of abstract geometry, which will constitute the cornerstone in the algebraization process that will be presented in the last section of the paper, we will need the following proposition (for the proof see Sections IV.5 and IV.6 of [3]): Proposition 2. Let hG, Si be a coordinated abstract geometry. Then, for all x, m, b ∈ R, there exists a unique y ∈ R, such that p(x, y) I l(m, b). Following Section IV.5 of [3], we introduce the notation I(x, m, b) to denote the unique y, such that p(x, y) I l(m, b), whose existence is postulated in the conclusion of Proposition 2. Note that for this notation to be meaningful, one must have available not only the abstract geometry G = hP, L, Ii, but also a chosen fixed coordinate system S = h(O, I, X, Y ), R, ρi on G, even though this is not made explicit in the notation. 4. The Institution AG of Abstract Geometry In this section, we construct an institution that formalizes the logical system of abstract (affine plane) geometry. In this institution, the sentences are constructed using the incidence of the abstract geometry and the models are coordinated affine planes, accompanied by intended interpretations of the signature variables in the coordinate spaces of the geometries. Satisfaction of an incidence formula by a model is defined based on whether the intended incidence relation denoted by the formula holds in the corresponding model 134 GEORGE VOUTSADAKIS under the stipulated interpretation of its variables in the coordinate space, which are, subsequently, translated to points and lines of the geometric model. Let X be a set. The set of terms Tm(X) over X is recursively defined as the smallest set, such that • X ⊆ Tm(X), and • T (t0 , s0 , s1 ) ∈ Tm(X), for all t0 , s0 , s1 ∈ Tm(X). The set of formulas Fm(X) over X, on the other hand, is the set Fm(X) = {(t0 , t1 ) I (s0 , s1 ) : t0 , t1 , s0 , s1 ∈ Tm(X)}. A function f : X → Tm(Y ) can be uniquely extended to f ∗ : Tm(X) → Tm(Y ) by setting • f ∗ (x) = f (x), for all x ∈ X, and • f ∗ (T (t0 , s0 , s1 )) = T (f ∗ (t0 ), f ∗ (s0 ), f ∗ (s1 )). Moreover, given such an f : X → Tm(Y ), we define f ∗ : Fm(X) → Fm(Y ) by f ∗ ((t0 , t1 ) I (s0 , s1 )) = (f ∗ (t0 ), f ∗ (t1 )) I (f ∗ (s0 ), f ∗ (s1 )), for all t0 , t1 , s0 , s1 ∈ Tm(X). Define Sign, the signature category of the institution under construction, as the category with objects all small sets and mappings f ∈ Sign(X, Y ) all set mappings f : X → Tm(Y ). Composition in this category is defined by setting, given f ∈ Sign(X, Y ) and g ∈ Sign(Y, Z), g ◦ f = g ∗ f : X → Tm(Z). Let SEN : Set → Set be the functor that maps a set X to the set of formulas Fm(X) and maps a function f : X → Tm(Y ) to its extension SEN(f ) = f ∗ : Fm(X) → Fm(Y ). Construct, next, the functor MOD : Sign → Catop as follows: Given X ∈ |Sign|, the category MOD(X) has objects all pairs of the form hhG, Si, σi, where hG, Si is a coordinated abstract geometry, i.e., G = hP, L, Ii is an abstract geometry and S = h(O, I, X, Y ), R, ρi is a coordinate system on G, and σ : X → R is a mapping from X to the underlying coordinate set R of S. Notice that such a mapping σ : X → R extends in a unique way to a mapping σ ∗ : Tm(X) → R as follows: 135 COORDINATIZATION IS ALGEBRAIZATION • σ ∗ (x) = σ(x), for all x ∈ X, and • σ ∗ (T (t0 , s0 , s1 )) = I(σ ∗ (t0 ), σ ∗ (s0 ), σ ∗ (s1 )), for all t0 , s0 , s1 ∈ Tm(X). The morphisms h : hhG, Si, σi → hhG′ , S ′ i, σ ′ i of MOD(X) are (strict coordinated) geometric morphisms h : hG, Si → hG′ , S ′ i, that make the following diagram commute: X σ R h @ @ σ′ @ R @ - R′ Given a mapping f : X → Y in Sign (i.e., f : X → Tm(Y ) in Set), the functor MOD(f ) : MOD(Y ) → MOD(X) is defined, for all hhG, Si, σi ∈ |MOD(Y )|, by MOD(f )(hhG, Si, σi) = hhG, Si, σ ∗ f i, and MOD(f )(h) = h : h(G, S), σ ∗ f i → h(G′ , S ′ ), σ ′∗ f i, for all h : hhG, Si, σi → hhG′ , S ′ i, σ ′ i in MOD(Y ). It is clear that this definition is sound, since the commutativity of the triangle displayed above implies the commutativity of X σ∗f R @ @ σ ′∗ f @ R @ - R′ h Finally, define, for all X ∈ |Set|, the relation |=X ⊆ |MOD(X)| × SEN(X), by setting, for all hhG, Si, σi ∈ |MOD(X)|, with G = hP, L, Ii, and all (t0 , t1 ) I (s0 , s1 ) ∈ SEN(X), hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ) iff σ ∗ (t1 ) = σ ∗ (T (t0 , s0 , s1 )). Let AG = hSign, SEN, MOD, |=i. AG is called the institution of abstract geometry. This terminology is justified by the following Theorem 3. The quadruple AG = hSign, SEN, MOD, |=i is an institution. 136 GEORGE VOUTSADAKIS Proof. Suppose f : X → Y be in Sign, (t0 , t1 ) I (s0 , s1 ) ∈ SEN(X) and hhG, Si, σi ∈ |MOD(Y )|, with G = hP, L, Ii. Then hhG, Si, σi |=Y iff iff iff iff iff iff SEN(f )((t0 , t1 ) I (s0 , s1 )) hhG, Si, σi |=Y (f ∗ (t0 ), f ∗ (t1 )) I (f ∗ (s0 ), f ∗ (s1 )) σ ∗ (f ∗ (t1 )) = σ ∗ (T (f ∗ (t0 ), f ∗ (s0 ), f ∗ (s1 ))) σ ∗ (f ∗ (t1 )) = I(σ ∗ (f ∗ (t0 )), σ ∗ (f ∗ (s0 )), σ ∗ (f ∗ (s1 ))) (σ ∗ f )∗ (t1 ) = I((σ ∗ f )∗ (t0 ), (σ ∗ f )∗ (s0 ), (σ ∗ f )∗ (s1 )) hhG, Si, σ ∗ f i |=X (t0 , t1 ) I (s0 , s1 ) MOD(f )(hhG, Si, σi) |=X (t0 , t1 ) I (s0 , s1 ). Thus, the satisfaction condition holds and AG is an institution. 5. The Ternary Ring R = hR, T, 0, 1i In this section, we review the abstract properties of the ternary ring R that is formed by the coordinatization of an abstract geometry G, as presented in Section IV.6 of [3]. Moreover, we describe the reverse process of coordinatization, presented in Section IV.7 of [3], by which an abstract geometry G is associated with a ternary ring. The abstract properties of the coordinatizing ternary rings as well as this construction of an abstract geometry using the elements and the algebraic properties of a ternary ring will prove useful when we study the process of algebraization of abstract geometry in the final sections of this paper. Let G = hP, L, Ii be an abstract geometry and let S = h(O, I, X, Y ), R, ρi be a coordinate system for G. Let T : R3 → R be the ternary operation on R defined by setting T (x, m, b) = I(x, m, b), for all x, m, b ∈ R. Then, as is shown in Section IV.6 of [3], the operation T on R satisfies the following properties: 1. T (0, m, b) = T (x, 0, b) = b, for all x, b, m ∈ R. 2. T (x, 1, 0) = T (1, x, 0) = x, for all x ∈ R. 3. The equation T (x, m, b) = T (x, m′ , b′ ) has a unique solution in R, for all m, m′ , b, b′ ∈ R, with m 6= m′ . 137 COORDINATIZATION IS ALGEBRAIZATION 4. The system of equations ( T (a, x, y) = b T (a′ , x, y) = b′ ) has a unique solution in R, for all a.a′ , b, b′ ∈ R, with a 6= a′ . 5. The equation T (a, m, x) = c has a unique solution in R, for all a, m, c ∈ R. The ternary ring associated with G and the coordinatization S of G will be denoted by R(G, S) or RS (G). More generally, an algebraic structure R = hR, T, 0, 1i, where T is a ternary operation satisfying properties 1-5 above, will be called a Hall ternary ring. Suppose, next, that a Hall ternary ring R is given. Then an abstract geometry G = hP, L, Ii may be constructed as follows: • P = {(x, y) : x, y ∈ R}; • L = {{(a, y) : y ∈ R} : a ∈ R}∪{{(x, T (x, m, b)) : x ∈ R} : m, b ∈ R}; • The incidence relation is simply the membership relation, i.e., for all (x, y) ∈ P and all l ∈ L, we have (x, y) I l iff (x, y) ∈ l. The following may now be established: Theorem 4. Given a Hall ternary ring, the structure G is an abstract geometry. Moreover, the Hall ternary ring associated with G under the coordinatization S = h((0, 0), (1, 1), (1, 0), (0, 1)), R, iR i coincides with the Hall ternary ring R. The coordinated abstract geometry that is obtained in this fashion out of the given Hall ternary ring R will be denoted by G(R). According to Theorem 4, we have that R(G(R)) = R. 6. The Institution GA of Geometric Algebra In this section, the algebraic institution GA, that will serve as the algebraic semantics of the institution AG of abstract geometry, will be constructed. Its models will be essentially Hall ternary rings and its sentences will be 138 GEORGE VOUTSADAKIS equations of terms over the language of Hall ternary rings. The fact that Hall ternary rings serve as coordinate rings for affine plane geometries is used in subsequent sections to construct mutually inverse interpretations between these two institutions. Let Sign be the category of signatures of the institution AG, defined in Section 4. Define the functor EQ : Sign → Set as follows: Given a set X, EQ(X) = {t0 ≈ t1 : t0 , t1 ∈ Tm(X)} and, given f : X → Tm(Y ) in Sign, define EQ(f )(t0 ≈ t1 ) = f ∗ (t0 ) ≈ f ∗ (t1 ), for all t0 , t1 ∈ Tm(X), where f ∗ : Tm(X) → Tm(Y ) is the unique extension of f on terms that was also defined in Section 4. Furthermore, define the functor ALG : Sign → Catop as follows: Given a set X, the category ALG(X) has as its objects all pairs hR, σi, where R is a Hall ternary ring and σ : X → R an assignment of elements from the universe R of R to the variables in X. A morphism h : hR, σi → hR′ , σ ′ i in ALG(X) is a Hall ternary ring homomorphism h : R → R′ , that makes the following diagram commute: X σ R @ @ σ′ @ R @ - R′ h Moreover, given an f : X → Tm(Y ) in Sign, the corresponding functor ALG(f ) : ALG(Y ) → ALG(X) sends an object hR, σi ∈ |ALG(Y )| to the object ALG(f )(hR, σi) = hR, σ ∗ f i and a morphism h : hR, σi → hR′ , σ ′ i in ALG(Y ) to ALG(f )(h) = h : hR, σ ∗ f i → hR′ , σ ′∗ f i in ALG(X). Finally, for every set X, define the satisfaction relation |=X ⊆ |ALG(X)| × EQ(X) by setting, for all hR, σi ∈ |ALG(X)| and all t0 , t1 ∈ Tm(X), hR, σi |=X t0 ≈ t1 iff σ ∗ (t0 ) = σ ∗ (t1 ). The quadruple GA = hSign, EQ, ALG, |=i is called the institution of geometric algebra, which is justified by the following: 139 COORDINATIZATION IS ALGEBRAIZATION Theorem 5. The quadruple GA = hSign, EQ, ALG, |=i is an institution. Proof. We check the satisfaction condition. Suppose that f : X → Tm(Y ) is a morphism in Sign, hR, σi ∈ |ALG(Y )| and t0 ≈ t1 ∈ EQ(X). Then, we have hR, σi |=Y EQ(f )(t0 ≈ t1 ) iff iff iff iff iff hR, σi |=Y f ∗ (t0 ) ≈ f ∗ (t1 ) σ ∗ (f ∗ (t0 )) = σ ∗ (f ∗ (t1 )) (σ ∗ f )∗ (t0 ) = (σ ∗ f )∗ (t1 ) hR, σ ∗ f i |=X t0 ≈ t1 ALG(f )(hR, σi) |=X t0 ≈ t1 . 7. GA is an Algebraic Semantics of AG In this section, it is shown that the institution GA of the equational logic of the Hall ternary rings, introduced in Section 6, constitutes an algebraic semantics of the institution AG of abstract geometry, introduced in Section 4. According to the theory of categorical abstract algebraic logic, an algebraic institution A = hSign′ , SEN′ , MOD′ , |=A i is an algebraic institution semantics of an institution I = hSign, SEN, MOD, |=I i if the corresponding π-institution π(I) = hSign, SEN, C I i is interpretable in π(A) = hSign′ , SEN′ , C A i. Note that C I is defined, for all Σ ∈ |Sign| and all Φ ∪ {φ} ⊆ SEN(Σ) by φ ∈ CΣI (Φ) iff, for every model M ∈ |MOD(Σ)|, M |=IΣ Φ implies M |=IΣ φ. A similar definition applies for C A , i.e., both π(I) and π(A) are the πinstitutions whose closure systems are the closure systems induced by the semantical entailment systems of the corresponding institutions. The πinstitution π(I) is interpretable in π(A) if there exists an interpretation hF, αi : I → A, i.e., a functor F : Sign → Sign′ and a natural transformation α : SEN → PSEN′ ◦ F , such that, for all Σ ∈ |Sign| and all Φ ∪ {φ} ⊆ SEN(Σ), φ ∈ CΣI (φ) iff αΣ (Φ) ⊆ CFA(Σ) (αΣ (φ)). 140 GEORGE VOUTSADAKIS To show that GA is an algebraic semantics of AG, we define the pair hISign , αi : AG → GA as follows: ISign : Sign → Sign is the identity functor on the common signature category of the two institutions. The natural transformation α : SEN → PEQ is defined by setting, for every set X and all t0 , t1 , s0 , s1 ∈ Tm(X), αΣ ((t0 , t1 ) I (s0 , s1 )) = {t1 ≈ T (t0 , s0 , s1 )}. Lemmas 6 and 7, that follow, are supporting lemmas for showing the main equivalence establishing the interpretation property of hISign , αi. This equivalence is shown in Proposition 8. The main theorem, Theorem 9, simply restates the equivalence of Proposition 8 in the language of abstract algebraic logic. Lemma 6. Let X be a set, hR, σi ∈ |ALG(X)| and t0 , t1 , s0 , s1 ∈ Tm(X). Then hR, σi |=X t1 ≈ T (t0 , s0 , s1 ) iff hG(R), σi |=X (t0 , t1 ) I (s0 , s1 ). Proof. hR, σi |=X t1 ≈ T (t0 , s0 , s1 ) iff iff iff iff σ ∗ (t1 ) = σ ∗ (T (t0 , s0 , s1 )) σ ∗ (t1 ) = I(σ ∗ (t0 ), σ ∗ (s0 ), σ ∗ (s1 )) (σ ∗ (t0 ), σ ∗ (t0 )) I (σ ∗ (s0 ), σ ∗ (s1 )) hG(R), σi |=X (t0 , t1 ) I (s0 , s1 ). Lemma 7. Let X be a set, hhG, Si, σi ∈ |MOD(X)| and t0 , t1 , s0 , s1 ∈ Tm(X). Then hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ) iff hRS (G), σi |=X t1 ≈ T (t0 , s0 , s1 ). Proof. hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ) iff σ ∗ (t1 ) = σ ∗ (T (t0 , s0 , s1 )) iff hRS (G), σi |=X t1 ≈ T (t0 , s0 , s1 ). COORDINATIZATION IS ALGEBRAIZATION 141 Proposition 8. Let X be a set. For all Φ ∪ {φ} ⊆ SEN(X), AG φ ∈ CX (Φ) iff GA αX (φ) ⊆ CX (αX (Φ)). AG (Φ). This means that, for all Proof. Assume, first, that φ ∈ CX hhG, Si, σi ∈ |MOD(X)|, hhG, Si, σi |=X Φ implies hhG, Si, σi |=X φ. Suppose, now, that hR, σi ∈ |ALG(X)|, and hR, σi |=X αX (Φ). Thus, hR, σi |=X t1 ≈ T (t0 , s0 , s1 ), for all (t0 , t1 ) I (s0 , s1 ) ∈ Φ. Therefore, by Lemma 6, we have hG(R), σi |=X (t0 , t1 ) I (s0 , s1 ), for all (t0 , t1 ) I (s0 , s1 ) ∈ Φ. Hence, by hypothesis, we get that hG(R), σi |=X φ. Again, using Lemma 6, we obtain that hR, σi |=X αX (φ). This proves that αX (φ) ⊆ GA (αΣ (Φ)). CX GA (αX (Φ)). This means that, Assume, conversely, that αX (φ) ⊆ CX for all hR, σi ∈ |ALG(X)|, hR, σi |=X αX (Φ) implies hR, σi |=X αX (φ). Suppose, now, that hhG, Si, σi ∈ |MOD(X)|, such that hhG, Si, σi |=X Φ. Thus, hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ), for all (t0 , t1 ) I (s0 , s1 ) ∈ Φ. Therefore, by Lemma 7, hRS (G), σi |=X t1 ≈ T (t0 , s0 , s1 ), for all (t0 , t1 ) I (s0 , s1 ) ∈ Φ. Hence, by hypothesis, we get that hRS (G), σi |=X αX (φ). Again, using Lemma 7, we obtain that hhG, Si, σi |=X φ. This AG (Φ). proves that φ ⊆ CX Theorem 9. The institution of Hall ternary rings GA is an algebraic semantics of the institution AG of abstract geometry. Proof. This is just a restatement of Proposition 8. 8. Algebraization of AG In this section, it is shown that not only is GA an algebraic semantics of AG, but, moreover, AG is an algebraizable institution with GA its algebraic counterpart. We do this by showing that there exists an interpretation hISign , βi from GA into AG, which is inverse to the interpretation hISign , αi. Thus, the two institutions AG and GA are deductively equivalent institutions, as is required for algebraizability. To this end, define the natural transformation β : EQ → PSEN, by setting, for every set X and all terms t0 , t1 ∈ Tm(X), βX (t0 ≈ t1 ) = {(t0 , t1 ) I (1, 0)}. 142 GEORGE VOUTSADAKIS Then we have the following analogs of Lemmas 6 and 7 and of Proposition 8 establishing that hISign , βi is an interpretation from GA to AG: Lemma 10. Let X be a set, hR, σi ∈ |ALG(X)| and t0 , t1 Tm(X). Then hR, σi |=X t0 ≈ t1 iff hG(R), σi |=X (t0 , t1 ) I (1, 0). Proof. hR, σi |=X t0 ≈ t1 iff iff iff iff σ ∗ (t0 ) = σ ∗ (t1 ) σ ∗ (t0 ) = I(σ ∗ (t1 ), 1, 0) σ ∗ (t0 ) = σ ∗ (T (t1 , 1, 0)) hG(R), σi |=X (t0 , t1 ) I (1, 0). Lemma 11. Let X be a set, hhG, Si, σi ∈ |MOD(X)| and t0 , t1 ∈ Tm(X). Then hhG, Si, σi |=X (t0 , t1 ) I (1, 0) iff hRS (G), σi |=X t0 ≈ t1 . hhG, Si, σi |=X (t0 , t1 ) I (1, 0) iff iff iff iff σ ∗ (t0 ) = σ ∗ (T (t1 , 1, 0)) σ ∗ (t0 ) = I(σ ∗ (t1 ), 1, 0) σ ∗ (t0 ) = σ ∗ (t1 ) hRS (G), σi |=X t0 ≈ t1 . Proof. Proposition 12. Let X be a set. For all E ∪ {t0 ≈ t1 } ⊆ EQ(X), GA t 0 ≈ t 1 ∈ CX (E) iff AG βX (t0 ≈ t1 ) ⊆ CX (βX (E)). GA (E). This means that, for Proof. Assume, first, that t0 ≈ t1 ∈ CX all hR, σi ∈ |ALG(X)|, hR, σi |=X E implies hR, σi |=X t0 ≈ t1 . Suppose, now, that hhG, Si, σi ∈ |MOD(X)|, and hhG, Si, σi |=X βX (E). Thus, hhG, Si, σi |=X (ǫ0 , ǫ1 ) I (1, 0), for all ǫ0 ≈ ǫ1 ∈ E. Therefore, by Lemma 11, hRS (G), σi |=X ǫ0 ≈ ǫ1 , for all ǫ0 ≈ ǫ1 ∈ E. Hence, by hypothesis, we have hRS (G), σi |=X t0 ≈ t1 . Again, using Lemma 11, we obtain that AG (βΣ (E)). hhG, Si, σi |=X βX (t0 ≈ t1 ). This proves that βX (t0 ≈ t1 ) ⊆ CX AG Assume, conversely, that βX (t0 ≈ T1 ) ⊆ CX (βX (E)). This means that, for all hhG, Si, σi ∈ |MOD(X)|, hhG, Si, σi |=X βX (E) implies hhG, Si, COORDINATIZATION IS ALGEBRAIZATION 143 σi |=X βX (t0 ≈ t1 ). Suppose, now, that hR, σi ∈ |ALG(X)|, such that hR, σi |=X E. Thus, hR, σi |=X ǫ0 ≈ ǫ1 , for all ǫ0 , ≈ ǫ1 ∈ E. Therefore, by Lemma 10, hG(R), σi |=X (ǫ0 , ǫ1 ) I (1, 0), for all ǫ0 ≈ ǫ1 ∈ E. Hence, by hypothesis, we get that hG(R), σi |=X βX (t0 ≈ t1 ). Again, using Lemma GA (E). 10, we obtain that hR, σi |=X t0 ≈ t1 . This proves that t0 ≈ t1 ∈ CX Theorem 13. The pair hISign , βi forms an interpretation from the institution GA of Hall ternary rings to the institution AG of abstract geometry. Proof. This is simply a restatement of Proposition 12. To complete our demonstration that AG is algebraizable and the institution of Hall ternary rings GA is its equivalent algebraic semantics, it suffices now to show that the two interpretations hISign , αi : AG → GA and hISign , βi : GA → AG are inverse of one another in the precise technical sense of [15] (see also [2]). In other words, it must be shown that composing α and β results in interderivable sets of geometric formulas in AG and composing β and α results in interderivable sets of equations in GA. The following two lemmas pave the way for the final results: Lemma 14. Let X be a set, hhG, Si, σi ∈ |MOD(X)| and (t0 , t1 ) I (s0 , s1 ) ∈ SEN(X). Then hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ) iff hhG, Si, σi |=X (t1 , T (t0 , s0 , s1 )) I (1, 0). Proof. hhG, Si, σi |=X (t0 , t1 ) I iff iff iff iff (s0 , s1 ) σ ∗ (t1 ) = σ ∗ (T (t0 , s0 , s1 )) σ ∗ (T (t0 , s0 , s1 )) = I(σ ∗ (t1 ), 1, 0) σ ∗ (T (t0 , s0 , s1 )) = σ ∗ (T (t1 , 1, 0)) hhG, Si, σi |=X (t1 , T (t0 , s0 , s1 )) I (1, 0). Lemma 15. Let X be a set, hR, σi ∈ |ALG(X)| and t0 , t1 ∈ Tm(X). Then hR, σi |=X t0 ≈ t1 iff hR, σi |=X t1 ≈ T (t0 , 1, 0). 144 GEORGE VOUTSADAKIS Proof. hR, σi |=X t0 ≈ t1 iff iff iff iff σ ∗ (t0 ) = σ ∗ (t1 ) σ ∗ (t1 ) = I(σ ∗ (t0 ), 1, 0) σ ∗ (t1 ) = σ ∗ (T (t0 , 1, 0)) hR, σi |=X t1 ≈ T (t0 , 1, 0). Theorem 16. The institution AG of abstract geometry and the institution GA of Hall ternary rings are deductively equivalent institutions. Thus, AG is algebraizable and the institution GA is its equivalent algebraic semantics. Proof. We have already proven in Theorems 9 and 13 that hISign , αi : AG → GA and hISign , βi : GA → AG are interpretations. Thus, it suffices to show that they are inverse to one another. Suppose, that X is a set and (t0 , t1 ) I (s0 , s1 ) ∈ SEN(X). Then, for all hhG, Si, σi ∈ |MOD(X)|, we have, by Lemma 14, that hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ) iff hhG, Si, σi |=X AG ((t0 , t1 ) I (s0 , s1 )) = βX (αX ((t0 , t1 ) I (s0 , s1 ))). Therefore, we get that CX AG CX (βX (αX ((t0 , t1 ) I (s0 , s1 )))). Similarly, if X is a set and t0 ≈ t1 ∈ EQ(X), then, for all hR, σi ∈ |ALG(X)|, we have, by Lemma 15, that hR, σi |=X t0 ≈ t1 iff hR, σi |=X αX (βX (t0 ≈ t1 )). Therefore, we get that GA GA (αX (βX (t0 ≈ t1 ))). This concludes the proof that (t0 ≈ t1 ) = CX CX hISign , αi and hISign , βi are indeed inverse to one another in the precise technical sense of abstract algebraic logic Theorem 16, which is the main theorem of the paper, shows that the institution of abstract (affine plane) geometry is deductively equivalent to the institution of geometric algebra. Thus, the class of all Hall ternary rings forms an equivalent algebraic semantics of affine plane geometry in the precise technical sense of abstract algebraic logic. It is in this sense that one may say that the coordinate rings of modern abstract geometry form an equivalent algebraic semantics of the logic of abstract geometry. 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